Chapter 14
Lifting line theory The simplest quantitative model used for predicting the aerodynamic force on a wing of finite span is the lifting line theory . It is cove covered red in most texts on aerodyna aerodynamic mics; s; e.g. e.g. Jones Jones (1942, (1942, ch. 5); Glauert (1926, ch. 10–11); Prandtl and Tietjens (1957, pp. 185–210); Abbott and von Doenhoff Doenhoff (1959, (1959, pp. 9–27); Milne-Tho Milne-Thomson mson (1973, ch. 10–11); Thwaites Thwaites (1987, pp. 305–311); Kuethe and Chow (1998, pp. 169–200); Anderson (2001, pp. 351–387); Bertin (2002, pp. 230– 256); and Moran (2003, pp. 135–138).
14.1 14 .1
Basi Basic c assum assumpt ptio ions ns of lifti lifting ng line line theo theory ry
Lifting line theory in its simplest form assumes that:
• the thickness and chord are much shorter than the span; • the wing is unswept; and • the flight is steady and perpendicular to the span. With the above assumptions, the wing is modelled (on the length scale of the span) as the line segment b −b x = y = 0, . (14.1) z 2 2 This line segment is called the lifting line .
14.2 14.2
The lifti lifting ng line, line, horses horseshoe hoe vort vortice ices, s, and the the wak wake e
If the wing is to be producing lift, the Kutta–Joukowsky theorem (3.66a) leads us to expect that there will be circulation around it in circuits lying in planes parallel to the plane of symmetry (i.e. planes of constant z ) and encircling the wing.
14.2.1 14.2.1
Deduc Deductio tions ns from from vorte vortex x theore theorems ms
Stokes’s theorem (§ 13.2.1) then requires that there are vortex filaments running along the wing. In the lifting line theory, these filaments are assumed to run along the lifting line (14.1). However, vortex filaments cannot end in the fluid ( § 13.2.2) so this description is physically incomplete incomplete.. Since, Since, by Helmholtz’s Helmholtz’s theorems, theorems, vortex vortex filaments filaments mov movee with the fluid, the simplest simplest 157
158
AERODYNAMICS I COURSE NOTES, 2006
consistent model is that wherever each filament would end on the lifting line, it instead trails behind the wing (back to a starting vortex at the take-off strip, or, essentially, to infinity).
14.2.2
Deductions from the wing pressure distribution
This picture of vortex filaments trailing behind the wing is consistent with the difference in the spanwise flow components over and under the wing induced by the pressure differences associated with the generation of lift (§ 12.1.2). When the upper and lower streams reunite at the trailing edge, they have the same speed (by Bernoulli’s equation, as the pressure is single-valued) and also the same u and v components (by the Kutta–Joukowsky condition); however, their spanwise components are different: inboard for the upper stream and outboard for the lower. Thus there forms a surface in the air behind the trailing edge across which the tangential (specifically spanwise) component of velocity is discontinuous. This is a vortex sheet , composed of vortex filaments, being the trailing legs of the vortex filaments inferred above from the vortex theorems.
14.2.3
The lifting line model of air flow
The model of the wing therefore consists of a collection of horseshoe vortices , each of which consists of a segment on the lifting line called a bound vortex (since it is constrained to move with the wing rather than allowed to drift in the flow) and two semi-infinite vortex filaments behind the wing called the trailing vortices . Together, the bound vortices of all the horseshoe vortices constitute the lifting line and represent the wing, and the collection of trailing vortices represent the wake . The flow around the wing is then taken as the sum of the contributions from the free stream q∞ = q ∞(i cos α + j sin α)
(14.2)
and the horseshoe vortices constituting the lifting line and the wake.
14.3
Horseshoe vortex
The horseshoe vortices are significant in so far as they affect the flow near the wing. The bound vortex part of the horseshoe doesn’t induce any velocity on the lifting line (on which it lies), but the trailing vortices do. Consider as an example, a horseshoe vortex filament of strength Γ running from ∞i + 2b k to b k to − 2b k to ∞i − 2b k , as illustrated in figure 14.1. The unit vectors of the three legs are −i , 2 −k , and +i , respectively. The strength, Γ , must be common to the three legs, by the vortex laws (§ 13.2.2). The velocity induced on the line of the bound vortex ( x = y = 0 ) can be calculated from two applications of the formula for points in the perpendicular plane of the end of a semi-infinite rectilinear vortex filament (13.28) q=
Γ ℓˆ × (r − r ′ ) . 4π |ℓˆ × (r − r′ )|2
Here, we can take ℓˆ = ∓i and r′ = ± 2b k for the two trailing vortices so that, for some point on the lifting line r = z k , b b ℓˆ × (r − r′ ) = ∓i × (z ∓ )k = ±(z ∓ ) j .
2
2
(14.3)
159
Lifting line theory
y
b
−
(0, 0,
2
)
x
z
(0, 0, +2b ) Figure 14.1: A rectangular horseshoe vortex, lying in the zx -plane with vertices at ± 2b k . Therefore, q (z k) =
= = =
Γ
4π Γ
4π Γ
4π
z−
b 2
b
|z − 2 |
2
−
z+
b 2
b
|z + 2 |2
(14.4a)
j
(z + 2b )2 (z − 2b ) − (z − 2b )2 (z + 2b ) j (z + 2b )2 (z − 2b )2
(14.4b)
(14.4c)
(z + 2b ) − (z − 2b ) j (z + 2b )(z − 2b )
−Γ j
π b 1 −
2z
2
;
(14.4d)
b
cf. Glauert (1926, p. 134), Kuethe and Chow (1998, p. 174), or Anderson (2001, p. 361). Notice that, having assumed that the trailing vortices lie in the zx -plane (the y = 0 plane), the velocity induced at the bound vortex by the trailing vortices is purely perpendicular to this plane; i.e. vertical. For |z | < 2b (i.e. on the bound vortex), v ≡ j · q < 0, so that this velocity is called downwash and denoted vw . The downwash is plotted in figure 14.2.
14.4
Continuous trailing vortex sheet
A difficulty with using a single horseshoe vortex to model a wing is that the downwash given by (14.4d) is infinite at z = ± 2b . This difficulty can be obviated by using a continuous trailing sheet of vortex filaments. If the trailing vortex filament from z ′ k to ∞i + z ′ k has strength γ (z ′ )δz ′ , it contributes γ (z ′ )δz ′ i × (z − z ′ )k δq (z k) = 4π (z − z ′ )2 −γ (z ′ ) j ′ = δz 4π (z − z ′ )
(14.5) (14.6)
to the downwash at r = z k. The total downwash vw is 1 vw (z ) ≡ j · q(z k) = − 4π
b/2
b/2
−
γ (z ′ ) dz ′ . z − z′
(14.7)
160
AERODYNAMICS I COURSE NOTES, 2006 2.0
1.5
1.0
) b / Γ ( /
0.5
w
v
, H S 0.0 A W N W O -0.5 D
-1.0
-1.5
-2.0 0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
SPANWISE POSITION, z/b
Figure 14.2: The downwash induced along the lifting line by a horseshoe vortex.
14.4.1
Circulation distribution and the wake
Since vortex filaments cannot end in the fluid, and since it is the strength of the bound vortex filaments that makes up the circulation around a section of the wing, the strength of the trailing vortex filament arriving at the lifting line at z = z ′ must be related to the circulation there by Γ (z ′ ) + γ (z ′ ) δz ′ = Γ (z ′ + γ (z ′ ) ∼
δz
′
) ∼ Γ (z ′ ) +
dΓ ′ δz dz ′
dΓ ; dz ′
(14.8a) (14.8b)
as in figure 14.3. Thus, in terms of the circulation distribution, the downwash is
−1 vw (z ) = 4π
b/2
b/2
−
dΓ dz ′ ; dz ′ z − z ′
(14.9)
cf. Glauert (1926, p. 135), Prandtl and Tietjens (1957, p. 199), Kuethe and Chow (1998, p. 174), or Anderson (2001, p. 364).
14.4.2
The form of the wake
In lifting line theory, the wake is assumed to be confined to a plane strip behind the wing. In fact, the situation is not so simple. We have seen (§ 14.3) that the horseshoe vortex filaments constituting the wake generate a downwash inside the horseshoe. Outside the horseshoe, they generate an upwash. Since vortex filaments move with the fluid, the effect of this is that the outer filaments of the sheet drift upward and the edges of the sheet curl inward. These curled edges form what are called the two wing-tip vortices , and are clearly visible when there is dust in the air; e.g. in crop-spraying (Kuethe and Chow 1998, p. 172, figure 6.5). They are also made
161
Lifting line theory
Γ(z ′ ) γ (z ′ ) δz ′
Γ(z ′ + δz ′ ) Figure 14.3: Relation between the strength of the trailing vortex filaments and the distribution of circulation along the lifting line. visible very commonly because, being regions of concentrated vorticity and therefore high speed and, via Bernoulli’s equation, low pressure, they cause atmospheric moisture to condense into liquid droplets, as in figure 7.3 of Bertin (2002, pp. 231–233). In spite of this very visible phenomenon, the model of a flat wake is still appropriate for computing the downwash on the wing. This is because the curling occurs while the air is passing downstream behind the wing, so that immediately behind the wing the wake is flatter than implied by the photographs. From the Biot–Savart law ( § 13.5.1), the importance of a piece of a vortex filament decreases like the inverse square of the distance; therefore, it is the nearer part of the wake that is more important.
14.5
The effect of downwash
We have established that the system of trailing vortex filaments in the wake cause a downwash at the wing. What effect does this have on the aerodynamics? Basically, instead of experiencing the free-stream velocity (14.2), the section at r = z k experiences that plus the downwash; i.e. q (z ) = (q ∞ cos α)i + {q ∞ sin α + vw (z )} j .
14.5.1
(14.10)
Effect on the angle of incidence: induced incidence
One effect is that instead of experiencing the geometric angle of incidence α ≡ arctan
q ∞ sin α = α, q ∞ cos α
(14.11)
162
AERODYNAMICS I COURSE NOTES, 2006
the section experiences α′ (z ) = arctan
q ∞ sin α + vw (z ) , q ∞ cos α
(14.12)
which is called the effective incidence (Anderson 2001, p. 354). For small angles (tan α ∼ α) and small downwash (vw ≪ q ∞) this reduces to α′ (z ) ∼ α +
vw (z ) . q ∞
(14.13)
The angle αi (z ) ≡ α − α′ (z ) =
−vw (z ) q ∞
(14.14)
is called the induced incidence (Anderson 2001, p. 365). Equation (14.14) may be combined with (14.9) to give αi (z ) =
−1 4π q ∞
b/2
b/2
−
dΓ dz ′ ; dz ′ z ′ − z
(14.15)
cf. Anderson (2001, p. 365, equation 5.18).
14.5.2
Effect on the aerodynamic force: induced drag
In lifting line theory, we assume that the aerodynamics of each two-dimensional section is basically the same as in two dimensions, except that the section experiences the free stream modified by the addition of the downwash. Since the free stream is rotated by the induced incidence αi , the lift and drag components of the aerodynamic force per unit span are altered in accordance with (1.3)–(1.4) to C ℓ′ = C ℓ cos αi − C d sin αi ′
C d = C ℓ sin αi + C d cos αi ,
(14.16a) (14.16b)
where the unprimed coefficients refer to two-dimensional conditions and the primed to threedimensional. For perfect fluid flow C d = 0 , and even when viscous effects are accounted for, C d ≪ C l for a functioning wing section below stall incidence (Jones 1942, p. 85). With the small angle approximation for αi , C ℓ′ = C ℓ ′
C d = C ℓ αi .
(14.17a) (14.17b)
Essentially, since the effective free stream has been rotated downwards and the aerodynamic force acts, in accordance with the Kutta–Joukowsky theorem, at right-angles to this, some of the two-dimensional lift force has been rotated backwards and acts as a drag; this drag is called the induced drag .
14.6
The lifting line equation
In (14.15), we can substitute for the local circulation using the Kutta–Joukowsky theorem (3.66): Γ =
ℓ(z ) 1 = q ∞c(z )C ℓ (z ) ρq ∞ 2
(14.18)
163
Lifting line theory
1/2
θ 0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
SPANWISE POSITION, z/b
Figure 14.4: A construction of the eccentric angle (14.20). to get Prandtl’s lifting line equation (Abbott and von Doenhoff 1959, p. 20, equation 1.10) 1 αi (z ) = − 8π
14.7
b/2
b/2
−
d(cC ℓ ) dz ′ . dz ′ z ′ − z
(14.19)
Glauert’s solution of the lifting line equation
The lifting line equation (14.19) is more easily solved by changing the variable integration from the spanwise coordinate z to the eccentric angle θ via θ ≡ arccos
−2z b
(14.20a)
−b cos θ (14.20b) 2 illustrated in figures 14.4–14.5 The lifting line equation (14.19) in terms of the eccentric angle is z≡
αi =
−1 π
cC ℓ 4b
π
d
dθ′
0
dθ′ . cos θ − cos θ′
(14.21)
Comparing this with Glauert’s integral (5.43) π
0
cos nθ′ dθ ′ −π sin nθ , = ′ cos θ − cos θ sin θ
(14.22)
we see that if we can expand the spanwise lift loading as (cf. Abbott and von Doenhoff 1959, p. 22) cC ℓ = 4b
∞
n=1
An sin nθ
(14.23)
164
AERODYNAMICS I COURSE NOTES, 2006
3
θ
, 2 E L G N A C I R T N E C C E 1
0 0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
SPANWISE POSITION, z/b
Figure 14.5: The eccentric angle (14.20) used to described spanwise lift loadings.
so that ∞
d cC ℓ nAn cos nθ , = dθ 4b n=1
(14.24)
then the lifting line equation (14.21) gives the induced incidence as (cf. Abbott and von Doenhoff 1959, p. 22)
αi =
π
−1
dθ′ nAn cos nθ cos θ − cos θ ′ n=1
π
0
∞
=
nAn
n=1 ∞
=
∞
n=1
π
−1 π
0
′
cos nθ′ dθ′ cos θ − cos θ′
nAn sin nθ . sin θ
(14.25a) (14.25b) (14.25c)
Note that the lift loading (14.23) goes to zero at the wing tips z = ± 2b (i.e. θ = 0 and π ) , which makes sense since the pressure difference has to vanish there. Any function over − 2b < z < + 2b could be represented by a trigonometric series like (14.23) but with cosines as well as sines, but the cosines wouldn’t vanish at the tips and so the coefficients of the cosine terms would have to be zero for a function vanishing at the tips. Note also that if the lift loading is symmetric, An must be zero for all even n .
165
Lifting line theory
14.7.1
Wing properties in terms of Glauert’s expansion
Lift
Assuming we were able to determine Glauert’s expansion coefficients An in (14.23), we could compute the wing’s lift coefficient from
C L ≡
b/2 b/2
ℓ dz
−
1 2
(14.26a)
2 ρq ∞ bc¯
1 = bc¯ 4b = c 2¯
b/2
cC ℓ dz
π
0
cC ℓ dθ 4b
∞
= 2A
(14.26b)
b/2
−
π
An
n=0
(14.26c) sin nθ sin θ dθ .
(14.26d)
0
Using the trigonometric integrals (for integer m and n )
π
0
sin mθ sin nθ dθ =
,
m = n;
0,
m = n,
π
2
(14.27)
the lift coefficient is simply
C L = π AA1 .
(14.28)
Rolling moment
Similarly, the lift distribution (14.23) could be used to compute the rolling moment (the component of the moment about the x-axis); of course, this vanishes when the lift distribution is symmetric:
c(−z )C ℓ (−z ) = c(z )C ℓ (z ) .
(14.29)
166
AERODYNAMICS I COURSE NOTES, 2006
Induced drag coefficient
Using (14.17b) for the sectional induced drag coefficient, the wing’s induced drag coefficient is C D =
= = =
D 1 2
b/2 b/2
d dz
−
1 2
(14.30b)
2 ρq ∞ bc¯
b/2 b/2
1 2
−
1 2
b/2 b/2
1 = bc¯
2 ρq ∞ cC d dz
(14.30c)
2 ρq ∞ bc¯
cC d dz
−
4 = c¯
(14.30a)
2 ρq ∞ bc¯
(14.30d)
bc¯ b/2
cC l αi dz
(14.30e)
cC l αi dz 4b
(14.30f)
b/2
−
b/2
b/2
−
π
= 2A
0
cC l αi sin θ dθ 4b
π
= 2A
0
m
n
sin θ dθ
sin mθ sin nθ dθ
π
(14.30i) (14.30j)
2
nA2n .
= π A
(14.30h)
0
n
nA2n
= 2A
sin mθ mAm sin θ
π
mAm An
m
An sin nθ
n
= 2A
(14.30g)
(14.30k)
n
Thus only the first term of Glauert’s expansion (14.23) contributes to the lift, but all terms induce drag; moreover, each term induces a positive amount of drag since squares are nonnegative.
14.8
The elliptic lift loading
Solving the lifting line equation for the lift distribution is quite difficult, but an easier problem is the investigation of a given lift distribution. Since only the first term in Glauert’s expansion (14.23) contributes to the lift, let’s examine first the lift distribution just consisting of this term: 2 2z cC ℓ C L C L = A1 sin θ = sin θ = 1− (14.31) , b 4b π A π A
or in dimensional terms ℓ=
4L π b
1−
2z b
2
,
as plotted in figure 14.6. It is called the elliptic lift loading .
(14.32)
167
Lifting line theory
1.0
0.8
) 0.6 0 ( l / ) z ( l
, T F I L 0.4
0.2
0.0 0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
SPANWISE POSITION, z/b
Figure 14.6: Elliptic lift loading (14.32). The corresponding induced incidence is given by the lifting line equation (14.25) as αi =
C L , π A
(14.33)
which is uniform across the span (Prandtl 1921, p. 191; Glauert 1926, p. 143; Abbott and von Doenhoff 1959, p. 8; Milne-Thomson 1973, p. 202; Kuethe and Chow 1998, p. 179; Anderson 2001, p. 369; Bertin 2002, p. 243). The elliptic lift loading is clearly the only lift loading with this property.
14.9
Properties of the elliptic lift loading
By (14.30k), an elliptically loaded wing’s induced drag coefficient is 2 1
C D = π AA = π A
C L π A
2
=
2 C L , π A
(14.34)
(Prandtl 1921, p. 192; Glauert 1926, p. 143; Abbott and von Doenhoff 1959, p. 8; Milne-Thomson 1973, p. 203; Kuethe and Chow 1998, p. 179; Anderson 2001, p. 369; Bertin 2002, p. 241; Moran 2003, p. 142). ‘Drag polars’ of relation (14.34) are plotted in figure 14.7.
14.9.1
Same lift coefficient, different aspect ratio
If we take two elliptically loaded wings with different aspect ratios (A1 and A2 ) and vary their geometric incidences (α1 and α2 ) so that they have the same lift coefficients C L , the induced incidences αi,1 and αi,2 must be related by αi,1 − αi,2 =
C L π
1 A1
−
1 A2
.
(14.35)
168
AERODYNAMICS I COURSE NOTES, 2006 1.4
1.2 A=7
1.0
L
C
, T N E I C I F F E O C T F I L
0.8 A=1
0.6
0.4
0.2
0.0
-0.2
-0.4 0.0
0.1
0.2
INDUCED DRAG COEFFICIENT, C D
Figure 14.7: Theoretical drag–lift polars for elliptic lift loading and A = 1, 2, 3, . . . , 7 . N.B.: drag includes only induced drag, not profile drag (skin friction and form drag); cf. Prandtl (1921, figure 46, p. 193). This follows by forming (14.33) for each wing and subtracting the two equations. Similarly, on forming (14.34) and subtracting, the induced drag coefficients must be related by C D,1 − C D,2 =
2 C L
π
1 A1
−
1 A2
(14.36)
.
These two equations, given by Prandtl (1921, p. 194), Glauert (1926, p. 144), Abbott and von Doenhoff (1959, p. 8), Milne-Thomson (1973, p. 204), Kuethe and Chow (1998, pp. 182–183), and Bertin (2002, pp. 244, 242), are extremely useful: they allow the prediction of the properties of a wing with one aspect ratio from measurements or computations of the properties at another aspect ratio.
14.9.2
Elliptic lift loading minimizes induced drag
The induced drag coefficient for the general lift loading (14.30k) can be expressed as ∞
C D = π A
n=1
where
∞
nA2n = π AA21
2
n
n=1
An A1
∞
= C D,ell 1 +
n
n=2
An A1
C D,ell = π AA21
2
,
(14.37)
(14.38)
is the induced drag coefficient of a wing with the same lift coefficient but with elliptic lift loading. Since the sum contains only nonnegative terms and vanishes for elliptic loading (for which A1 is the only nonzero sine coefficient), a most important result follows: The induced drag coefficient is a minimum for elliptic loading.
169
Lifting line theory
14.10
Lift–incidence relation
Say we know the lift–incidence relation for infinite aspect ratio: C L,∞ = f (α∞) .
(14.39)
Then for a finite aspect ratio, A, with elliptic loading the induced incidence is αi =
C L , π A
(14.40)
and at geometric incidence α the lift coefficient is that for the infinite aspect ratio at geometric incidence α∞ = α − αi . (14.41) Since, for elliptic loading, the downwash and induced incidence are uniform along the span, if the wing is untwisted, the sectional lift coefficient C ℓ will be too and the total lift coefficient C L will change proportionately.
14.10.1
Linear lift–incidence relation
If the infinite aspect ratio (two-dimensional) lift–incidence relation is linear, C L = m(α − αi − α0 ) ,
(14.42)
but if the induced incidence is given by (14.33) then C L =
m
1+
m A
(α − α0 ) ;
(14.43)
π
thus, if the two-dimensional lift–incidence slope is m , the finite aspect ratio slope is dC L m = . dα 1 + mA
(14.44)
π
For 0 A ∞, this is less than m. Note also in (14.43) that the zero-lift incidence α0 is independent of aspect ratio. The result (14.43) for a thin aerofoil (m = 2π ) is illustrated in figure 14.8.
14.11
Realizing the elliptic lift loading
If the effective wing section lift coefficient is the same function of the effective incidence across the span (i.e. no twist), C ℓ is uniform too: C ℓ = C L .
(14.45)
For this to be compatible with the elliptical lift loading, since c(z ) ≡
ℓ(z ) 1 2
2 ρq ∞ C ℓ
(14.46)
,
the chord c(z ) must vary like ℓ(z ) =
2
2
ρq ∞c¯C L
π
1−
2z b
2
;
(14.47)
170
AERODYNAMICS I COURSE NOTES, 2006 1.4
1.2 A=7 1.0
0.8
L
C
, T N E I C I F F E O C T F I L
0.6
0.4 A=1 0.2
0.0 o
o
-10
0
10
o
20
o
-0.2
-0.4 GEOMETRIC ANGLE OF INCIDENCE,
α
Figure 14.8: Theoretical lift–incidence relations for thin wings with zero-lift incidence α0 = −5◦ and aspect ratio A = 1 , 2, 3, . . . , 7 ; cf. Prandtl (1921, figure 47, p. 193). i.e. elliptically
c(z ) 4 = c¯ π
1−
2z b
2
;
(14.48)
cf. Kuethe and Chow (1998, p. 178), Anderson (2001, p. 370), or Moran (2003, p. 143).
14.11.1
Corrections to the elliptic loading approximation
Practical wings are rarely constructed with an elliptic variation of chord length, since this is more expensive to manufacture than rectangular or trapezoidal planforms (Anderson 2001, p. 374). Nevertheless, the simple results for elliptic loading are appealing and useful at least as a first approximation. They are frequently used in a generalized form, with correction factors. For example, in place of the elliptic loading induced drag coefficient relation (14.34), Abbott and von Doenhoff (1959, p. 16) recommend a corrected formula which for untwisted unswept wings reduces to C D =
2 C L π Au
(14.49)
where u is an ‘induced-drag factor’ which depends on the taper ratio and aspect ratio. The values of u may be read off the charts of Abbott and von Doenhoff (1959, figure 10, p. 17). An alternative induced-drag correction factor is given by (Anderson 2001, p. 376). It should be noted that these correction factors don’t change the induced drag coefficient by more than about ten per cent over the practical range of taper ratio and aspect ratio (Anderson 2001, p. 376), so that even the uncorrected formulae are sometimes useful for obtaining rough initial estimates.
